Complexity: Between Order and Chaos

Is a human more complex than a water flea? If counting number of genes is your measure of complexity, the water flea is more complex. (Image of Daphnia from Wikipedia.)

How does one measure complexity? Intuitively, we feel that a more complex entity has more of ‘something’ than a less complex entity. What is that ‘something’? That’s more difficult to define.

In a 2013 paper,¹ three authors take on this problem in a paper titled What is a complex system? In the abstract, they write that “there is no concise definition of a complex system, let alone a definition on which all scientists agree.” Their paper reviews various methods used to characterize complex systems and enumerate complexity; they think the best so far is “Statistical Complexity”. This measure meshes with the idea that complexity lies between two poles: maximal order and ‘pure’ randomness.

A maximally ordered object is potentially “easy” to describe. In my thermodynamics classes, students learn the third law of thermodynamics: A perfect crystal at zero kelvin has an entropy of zero. There is only one possible arrangement for this perfect crystal. Yes, you can imagine it in your mind’s eye – rows upon rows of ordered atoms. On the other hand, picture the atoms of an ideal gas moving around in a container at some decent temperature. The movement is chaotic, especially if you try to follow one of these atoms. It zigs and zags in multiple collisions which are somewhat dependent on what all the other atoms are doing. A 24-liter container has roughly a mole of gas particles, that’s six gazillion of them (or Avogadro’s number). Keeping track of individual atoms is a nightmare, but we can easily describe their macroscopic behavior with seemingly simple quantities such as pressure and temperature – easily enumerated and seemingly ‘constant’ at least for a large number of particles. Liquids are much harder to describe than either solids or gases. Perhaps they are more complex?

Somewhere in between those two extremes, order and chaos, is where all the interesting stuff happens. Living systems, in particular, aren’t perfectly ordered systems nor are they just a random grab-bag of chemicals. Just throwing together all the molecules does nothing. They just sit. Dead. There’s something about living systems. that seems complicated and not so easily described. They’re complex! Perhaps the question “What is Life?” should be replaced with “What is Living?”

The 2013 paper throws up some considerations of complex systems. Certain common words or phrases get bandied about regularly in such discussions. Nonlinearity. Feedback loops. Robustness. Irreversibility. Non-equilibrium. Emergence. Multi-layered organization. Information. Memory. Probability. There are several equations typically involving sums and logarithms. I admit to not understanding chunks of the paper where my eyes glazed over. One phrase jumped out at me, though: coarse-graining.

Coarse-graining is what computational scientists like me do when we are trying to model larger more complex systems. We strip out microscopic features that don’t seem to make a large contribution so we can simplify the system while hopefully retaining the essentials at the length scale being studied. For example, if I was interested in the dynamics of protein motion, I might ignore protons, neutrons and electrons in atoms and chemical bonds and replace them by balls connected together with Hooke’s Law springs. That allows me to use the more tractable equations of classical mechanics rather than quantum mechanics. But if I’m interested of how that protein interacts with other molecules in a solution, I might replace the balls and springs with a blob that captures the overall electrostatic map of the protein. Yet again I have replaced a fine-grained model with a coarser-grained one.

It turns out that coarse-graining might be an approach to study (or even explain!) how complexity arises from simpler systems. I study the origin of life, perhaps the prime example that illustrates this thorny question. So I read another interesting paper from 2017 titled Coarse-graining as a downward causation mechanism.² The crux of the idea is that a certain robustness can be achieved when some aggregate measure of the microscopic properties allows for predicting the future state of the system. The trick here is that different parameters/variables are important on different length and time scales, and a system can build complexity when over time, there is an adaptation to a more robust state. A seemingly homogenous system now becomes a two-layer system allowing for some separation of variables. Further coarse-graining can then lead to a multi-layered hierarchy, possibly even with control elements.

Both the papers are mainly about ideas. While examples are drawn from multiple areas to illustrate the idea, we don’t exactly know how to measure complexity. Complex systems, are, for lack of a better word, complex! The idea of complexity being somewhere between order and chaos, and the adaptive approach of coarse-graining, may help us in the difficult task of understanding such systems. We do know that there are many, many ways to fail!

1. J. Ladyman, J. Lambert, K. Wiesner, K. Euro Jnl Phil Sci (2013) 3:33-67.

2. J. C. Flack, Phil. Trans. R. Soc. A (2017) 375:20160338.